Estimating the unknown parameters and states of a dynamical system can be achieved through MHE. This framework reformulates the state estimation problem as an optimization problem dual to MPC [1]}. This work assumes full state observability, with the realized discrete-time dynamics given by
\(\begin{split}\mathbf {x}_{t + 1} & = \mathbf {F}(\mathbf {x}_t, \mathbf {u}_t; \theta ^*) + \mathbf {w}_t , \\\mathbf {x}_1 & \sim \mathcal {N}(\hat{\mathbf {x}}_1, \Sigma _{\mathbf {x}_1}) ,\end{split}\) 
